Tropical Ptolemy Transformations and Invariants of Braids

We present a new construction of invariants for spherical braids using tropical geometry. Given a braid on \( n \geq 5 \) strands on the 2-sphere, we associate to it a sequence of Delaunay triangulations connected by edge flips. Each triangulation carries edge labels valued in a tropical semifield, and each flip updates the labels via the tropical Ptolemy relation: \[ x \oplus y = (a \oplus c) \otimes (b \oplus d), \quad where \oplus = \max, \otimes = +. \] This process respects flip identities such as involution, far-commutativity, and the pentagon relation. We show that the resulting label at the end of the sequence defines an invariant of the braid up to isotopy. This construction offers a combinatorial framework for studying braid groups through tropical methods and enriches the connection between low-dimensional topology and tropical geometry.